Russell’s Other Contradiction: the Paradox of Propositions

Appendix B of The Principles of Mathematics (1903) contains Russell’s first statement of his theory of logical types. This simple version of the theory is designed to block the reasoning that leads to the paradox of the Russell class. But Russell notes immediately that new problems arise. The problems culminate in the paradox of propositions. This is a problem that seems to run in exact parallel to the paradox of the Russell class. It seemed therefore desirable to Russell that a single solution to both paradoxes be found. Since the simple theory of types (ST) does not offer such a solution it is commonly believed that the paradox of propositions was Russell’s principal motive— at least at the time when he had just finished writing the Principles—for searching for and eventually formulating a ramified theory of types (RT). In the next section I shall present the very first version of ST as it occurs in the Principles. I shall explain in which direction Russell was seeking for a solution to the paradox of propositions which would run in parallel to his solution to the class paradox. Next I turn to the Russell-Frege correspondence of 1902 and 1903. Apart from Appendix B this is the only place in Russell’s writings in which the paradox is mentioned. It turns out that the paradox was one of the principal topics in the correspondence between Frege and Russell during these years. I shall reconstruct Frege’s suggestion as to how the paradox may be resolved and show that it is unsuccessful. Finally I shall describe Russell’s own, rather unspectacular solution. In conclusion I shall argue that although RT can be used to block the paradox of propositions, this is is not the solution Russell came to adopt. Russell found a simple solution to the paradox some years before he formulated RT. There is no evidence or other reason to believe that Russell changed his mind as to the proper solution of the paradox of propositions after he discovered RT. Instead there are good reasons to support the view that Russell remained content with the solution he had found in correspondence with Frege.